Eccentricity

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In mathematics, the eccentricity (sometimes spelled "excentricity"), denoted
ε (or, for basic text notation "e"), is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

Contents

Definitions

Where P is any point on the curve, a conic section is defined by the condition that the distances between P and a fixed point F and between P and a fixed line L are in the ratio ε:1 (ε > 0). F is termed the focus of the conic section, L the directrix, and ε the eccentricity.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis being vertical, the eccentricity is

where α is the angle between the plane and the horizontal and β is the angle between the cone and the horizontal.

The linear eccentricity of a conic section, denoted c or E, is the distance between its center and its focus (or one of its two foci).

Alternative names

The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called numerical eccentricity.

In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation.

We define a number of related additional concepts (only for ellipses):

name

symbol

value in terms of and

value in terms of

angular eccentricity

first eccentricity

second eccentricity

third eccentricity

Quadrics

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest ("major") and the shortest ("minor") axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

Celestial mechanics

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipse, in Keplerian, i.e., potentials.